We propose a construction of rate-compatible punctured codes based on protographs that have a special E2RC structure in their parity part (E2RC codes were introduced in Kim, Ramamoorthy and McLaughlin '06) . The protograph representation of these codes facilitates their asymptotic performance analysis and allows the implementation of high speed decoders. The construction process starts with a good high rate protograph. The protographs of lower rate codes are derived from the higher rate protographs via the process of check-splitting. The check-splitting is done in a specific manner so that the parity nodes in the protograph have the special E2RC structure. We also present additional design rules that ensure that the gap to capacity remains low across the range of rates. Using our approach we exhibit protographs that have a gap of at most 0.27 dB to capacity across the range of rates 1/2 to 8/9. These conclusions are supported by our simulation results. Our work, therefore presents a systematic method for the design of E2RC-like codes.